# All Questions

1,271,775 questions
Filter by
Sorted by
Tagged with
4 views

9 views

### Combinatorics, sequences

What is the number of non-increasing sequences of length n with elements from {0,1,2,⋯,n} with fixed first element k, k=0, 1, ... , n ? Consider the set of numbers from 1 to n. We construct non-...
16 views

4 views

22 views

### Find a basis for S $\cap$ T. Also find its dimension. Conditions are as following.

Let $S=\left\{\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\in\Bbb{R}:a_{1}+a_{2}+a_{3}+a_{4}=0\right\}$ And $T=\left\{\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\in\Bbb{R}:a_{1}-a_{2}+a_{3}-a_{4}=0\right\}$ ...
16 views

### Are there more learnable but undecidable cases except the halting problem

In the ICML 1992 paper, On the Learnability of the Uncomputables, by Richard Lathrop, he proved that halting problem is learnable in a probabilistic learnability. So except halting problem, are there ...
29 views

### Showing a function is constant - Complex analysis

I am trying to solve the following problem. $f(z)=u(x,y)+iv(x,y)$ is an analytic function in $D$ ($D$ is connected and open). If $u, v$ fulfill the relation $G(u(x,y), v(x,y))= 0$ in $D$ for some ...
11 views

### Domain of the natural logarithm of a factorisable quadratic polynomial

Suppose I have a function that is the natural logarithm of a quadratic polynomial, which can be factorised: $$f(x) = \ln(x^2+2x-8)$$ The domain of $f$ is $x \lt -4$ and $x \gt 2$. However, if I ...
8 views

### Gradient and Hessian of squared Frobenius norm

I want to find the Gradeint and Hessian of the following function, $F(\mathbf{S}) = \frac{1}{2}\Vert \mathbf{M} - \mathbf{K_2SK_1^T}\Vert _F^2+\frac{1}{2}\Vert\mathbf{S}\Vert_F^2$. My try: Using trace ...
38 views

### What are some other ways of solving this limit?

I have come across many limit questions of the form $0^0$ For instance , here is one example $$y=\lim_{x \to 0^+} (2\sin(\sqrt x) + \sqrt x\sin{1\over x})^x$$ In this example if I take logarithm on ...
12 views

### Help with finding lower bound of a binomial expansion

I have to find the lower bound of $1 - [1 - (2^{-m/2} - 2^{-m})]^p$. I should get the answer as $p*2^{-m/2} - p(p+1)*2^{-m}$. I tried to do binomial expansion but I am stuck. Any help is appreciated.
11 views

### Question about $T-$cyclic subspace generated

$\textbf{Definition:}$ Let $V$ be a finite dimensional vector space over a field $F$ and let $T:V \to V$ be a linear operator. If $v$ is a vector in $V$, the $T-$cyclic subspace generated by $v$ is ...
14 views

### Expected value of red/blue ball game

There are 4 red balls and 3 blue balls in an urn. Picking red balls gives you 1 dollar and picking blue balls means you have to pay 1 dollar. You can stop picking balls at any point. What is the ...
17 views

### $[G:K]=[G:H][H:K]$ for arbitrary group in ZF

If $G$ is any group, and $H$ and $K$ are subgroups such that $H\supseteq K$, then can we prove that $[G:K]=[G:H][H:K]$ only using ZF, where $[A:B]$ is the cardinality of the set $\{Bx:x\in A\}$, where ...
My understanding is that functions of the form $f(g(x))$ can be differentiated using the "chain rule", where $$\frac{d}{dx}f(g(x)) = f'g(x) \cdot g'(x)$$ I was trying to apply that logic to ...